Exam Questions & Step-by-Step Markschemes (5 Problems)
Question 1 • 2023
6 Marks
Find the shortest distance between the two lines: L_1: \vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda (2\hat{i} + 3\hat{j} + 4\hat{k}) L_2: \vec{r} = (2\hat{i} + 4\hat{j} + 5\hat{k}) + \mu (3\hat{i} + 4\hat{j} + 5\hat{k})
Answer: d = \frac{1}{\sqrt{6}} = \frac{\sqrt{6}}{6}\text{ units}
Question 2 • 2022
4 Marks
Find the angle between the two planes \vec{r} \cdot (2\hat{i} - \hat{j} + 2\hat{k}) = 3 and \vec{r} \cdot (3\hat{i} + 6\hat{j} + 2\hat{k}) = 5 .
Answer: \theta = \arccos\left(\frac{4}{21}\right) \approx 79.02^\circ
Question 3 • 2024
4 Marks
Show that the four points with position vectors A(4\hat{i} + 5\hat{j} + \hat{k}) , B(-\hat{j} - \hat{k}) , C(3\hat{i} + 9\hat{j} + 4\hat{k}) , and D(-4\hat{i} + 4\hat{j} + 4\hat{k}) are coplanar.
Answer: [\vec{AB} \, \vec{AC} \, \vec{AD}] = 0 \implies \text{Points are coplanar}
Question 4 • 2024
6 Marks
Find the shortest distance between the skew lines whose vector equations are: \vec{r}_1 = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - 3\hat{j} + 2\hat{k}) \vec{r}_2 = (4\hat{i} + 5\hat{j} + 6\hat{k}) + \mu(2\hat{i} + 3\hat{j} + \hat{k})
Answer: d = \frac{3\sqrt{19}}{19} \text{ units}
Question 5 • 2024
4 Marks
Find the value of \lambda for which the four points with position vectors \vec{a} = 3\hat{i} - 2\hat{j} - \hat{k} , \vec{b} = 2\hat{i} + 3\hat{j} - 4\hat{k} , \vec{c} = -\hat{i} + \hat{j} + 2\hat{k} , and \vec{d} = 4\hat{i} + 5\hat{j} + \lambda\hat{k} are coplanar.
Answer: \lambda = -\frac{146}{17}